The production of four top quarks ($t\bar{t}t\bar{t}$) at hadron colliders constitutes one of the rarest Standard Model (SM) processes accessible at the Large Hadron Collider (LHC) and provides a uniquely sensitive probe of the top-quark Yukawa coupling, of its possible $CP$ structure, and of new physics encoded in four-heavy-fermion operators within the Standard Model Effective Field Theory (SMEFT) framework. We present a phenomenological study of $t\bar{t}t\bar{t}$ production in proton-proton collisions at $\sqrt{s}=13$~TeV, using a Hyper-Graph Neural Network (H-GNN) to discriminate multilepton signal events from the dominant SM backgrounds viz. $t\bar{t}W$, $t\bar{t}Z$, $t\bar{t}H$ and $t\bar{t}VV$ processes. In the H-GNN architecture each event is represented as a hypergraph whose nodes correspond to reconstructed jets and leptons and whose hyperedges encode higher-order correlations among arbitrary subsets of these objects, allowing the network to learn the many-body kinematic structures that characterize the $t\bar{t}t\bar{t}$ final state. Combining same-sign di-lepton, tri-lepton, and four-lepton channels following a CMS-like event selection, the H-GNN attains an area under the ROC curve of $0.97$ for the $t\bar{t}t\bar{t}$ signal and yields a statistical significance of $Z=6.8$ at $ℒ=140$~fb${}^{-1}$. We exploit the improved signal extraction to derive one-parameter $95\%$ confidence level limits on the Wilson coefficients of the dimension-six operators ${𝒪}_{\Phi u}$, ${𝒪}_{tt}^{(1)}$, ${𝒪}_{qq}^{(1)}$, ${𝒪}_{qt}^{(1)}$, and ${𝒪}_{qt}^{(8)}$. We project the expected sensitivity at the HL-LHC integrated luminosities of $1000{fb}^{-1}$ and $3000{fb}^{-1}$ with $50\%$ uncertainty on the background estimation. Finally, the sensitivity reach of H-GNN is compared to the SPANet and Particle Transformer baseline.